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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Tarski
Date: Fri, 7 Mar 2025 07:32:12 -0500
Organization: i2pn2 (i2pn.org)
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On 3/6/25 9:26 PM, olcott wrote:
> On 3/6/2025 6:36 AM, Richard Damon wrote:
>> On 3/5/25 7:36 PM, olcott wrote:
>>> On 3/5/2025 5:41 PM, Richard Damon wrote:
>>>> On 3/5/25 9:25 AM, olcott wrote:
>>>>> On 3/4/2025 10:59 PM, Richard Damon wrote:
>>>>>> On 3/4/25 9:25 PM, olcott wrote:
>>>>>>> On 3/4/2025 5:45 PM, Richard Damon wrote:
>>>>>>>> On 3/4/25 9:47 AM, olcott wrote:
>>>>>>>>> On 3/4/2025 6:29 AM, Richard Damon wrote:
>>>>>>>>>> On 3/3/25 10:24 PM, olcott wrote:
>>>>>>>>>>> On 3/3/2025 7:07 PM, Richard Damon wrote:
>>>>>>>>>>>> On 3/3/25 7:38 PM, olcott wrote:
>>>>>>>>>>>>> On 3/3/2025 9:05 AM, Mikko wrote:
>>>>>>>>>>>>>> On 2025-03-01 19:42:50 +0000, olcott said:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> On 3/1/2025 2:37 AM, Mikko wrote:
>>>>>>>>>>>>>>>> On 2025-02-28 21:58:34 +0000, olcott said:
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> On 2/28/2025 3:56 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>> On 2025-02-26 14:42:23 +0000, olcott said:
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> On 2/26/2025 3:12 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>> On 2025-02-25 21:07:31 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> On 2/25/2025 9:27 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>> On 2025-02-24 21:31:26 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> On 2/24/2025 2:51 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 2025-02-22 17:24:59 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> On 2/22/2025 3:12 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-02-21 23:22:23 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2/20/2025 3:01 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2025-02-18 13:50:22 +0000, olcott said:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> There is nothing like that in the following 
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> concrete example:
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> LP := ~True(LP)
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In other words you are saying the Prolog is 
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> incorrect
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> to reject the Liar Paradox.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Above translated to Prolog
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ?- LP = not(true(LP)).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> LP = not(true(LP)).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> According to Prolog rules LP = not(true(LP)) 
>>>>>>>>>>>>>>>>>>>>>>>>>>>> is permitted to fail.
>>>>>>>>>>>>>>>>>>>>>>>>>>>> If it succeeds the operations using LP may 
>>>>>>>>>>>>>>>>>>>>>>>>>>>> misbehave. A memory
>>>>>>>>>>>>>>>>>>>>>>>>>>>> leak is also possible.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>>>>>>>>>>>>>>>>>>>>>>>>>>>> false
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> This merely means that the result of 
>>>>>>>>>>>>>>>>>>>>>>>>>>>> unification would be that LP conains
>>>>>>>>>>>>>>>>>>>>>>>>>>>> itself. It could be a selmantically valid 
>>>>>>>>>>>>>>>>>>>>>>>>>>>> result but is not in the scope
>>>>>>>>>>>>>>>>>>>>>>>>>>>> of Prolog language.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> It does not mean that. You are wrong.
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> It does in the context where it was presented. 
>>>>>>>>>>>>>>>>>>>>>>>>>> More generally,
>>>>>>>>>>>>>>>>>>>>>>>>>> unify_with_occurs_check also fails if the 
>>>>>>>>>>>>>>>>>>>>>>>>>> arguments are not
>>>>>>>>>>>>>>>>>>>>>>>>>> unfiable. But this possibility is already 
>>>>>>>>>>>>>>>>>>>>>>>>>> excluded by their
>>>>>>>>>>>>>>>>>>>>>>>>>> successfull unification.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> IT CANNOT POSSIBLY BE SEMANTICALLY VALID
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Of course it is. Its semantics is well defined 
>>>>>>>>>>>>>>>>>>>>>>>> by the Prolog standard.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Go freaking read the Clocksin and Mellish.
>>>>>>>>>>>>>>>>>>>>>>> an "infinite term" means NOT SEMANTICALLY VALID.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> Prolog does not define any semantics other than 
>>>>>>>>>>>>>>>>>>>>>> the execution semantics
>>>>>>>>>>>>>>>>>>>>>> of a prolog program. Therefore no data structure 
>>>>>>>>>>>>>>>>>>>>>> has any own semantics.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> The result of the exectution of an instruction 
>>>>>>>>>>>>>>>>>>>>>> like LP == not(true(LP))
>>>>>>>>>>>>>>>>>>>>>> is not fully defined by the standard so we may say 
>>>>>>>>>>>>>>>>>>>>>> that that instruction
>>>>>>>>>>>>>>>>>>>>>> is semantically invalid.
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> When we ask for Prolog to determine whether an 
>>>>>>>>>>>>>>>>>>>>> expression
>>>>>>>>>>>>>>>>>>>>> in Prolog is true according to its facts and rules 
>>>>>>>>>>>>>>>>>>>>> and the
>>>>>>>>>>>>>>>>>>>>> evaluation of the expression gets stuck in an 
>>>>>>>>>>>>>>>>>>>>> infinite loop
>>>>>>>>>>>>>>>>>>>>> then this expression IS SEMANTICALLY INCORRECT.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Which is not done anywhere above.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> In other words you can't remember things that I said
>>>>>>>>>>>>>>>>>>> a few messages ago and I have to endlessly repeat 
>>>>>>>>>>>>>>>>>>> everything
>>>>>>>>>>>>>>>>>>> every time?
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Is this just an instance or your favorite sin? If not, 
>>>>>>>>>>>>>>>>>> what do you think
>>>>>>>>>>>>>>>>>> I didn't remember?
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> page 3 has the liar paradox and the Cloksin & Mellish 
>>>>>>>>>>>>>>>>>>> Quote
>>>>>>>>>>>>>>>>>>> https://www.researchgate.net/ 
>>>>>>>>>>>>>>>>>>> publication/350789898_Prolog_detects_and_rejects_pathological_self_reference_in_the_Godel_sentence
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> It just says that your prolog system is defective as 
>>>>>>>>>>>>>>>>>> it does not reject
>>>>>>>>>>>>>>>>>> your LP = not(true(LP)). The Prolog standard says that 
>>>>>>>>>>>>>>>>>> this operation may
>>>>>>>>>>>>>>>>>> but need not fail. It also cortectly says that
>>>>>>>>>>>>>>>>>>  LP = not(true(LP)), write(LP)
>>>>>>>>>>>>>>>>>> would not work.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> There is no "need not fail" Clocksin and Mellish says
>>>>>>>>>>>>>>>>> impossible to succeed (paraphrase).
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> No, that is not said. In a footnote they say that the 
>>>>>>>>>>>>>>>> behaviour is undefined,
>>>>>>>>>>>>>>>> i.e., an implementation may choose what to do. They do 
>>>>>>>>>>>>>>>> say that a typical
>>>>>>>>>>>>>>>> implementation does not fail, which implies "need not 
>>>>>>>>>>>>>>>> fail".
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> More precisely it says that there is a cycle in the
>>>>>>>>>>>>>>>>> directed graph of the evaluation sequence of the 
>>>>>>>>>>>>>>>>> expression.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Assuming that the unification does not fail.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> That you fail to understands that the following means this
>>>>>>>>>>>>>>>>> is your lack of understanding not my mistake.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> ?- LP = not(true(LP)).
>>>>>>>>>>>>>>>>> LP = not(true(LP)).
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> It means that the pariticular implementation you used 
>>>>>>>>>>>>>>>> exploited the
>>>>>>>>>>>>>>>> "need not fail" permission, producing a cycle in the 
>>>>>>>>>>>>>>>> data structure.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>>>>>>>>>>>>>>>> false.
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